A034754 Dirichlet convolution of 3^(n-1) with phi(n).

1, 4, 11, 32, 85, 260, 735, 2224, 6585, 19780, 59059, 177472, 531453, 1595076, 4783175, 14351168, 43046737, 129147252, 387420507, 1162281440, 3486785925, 10460412292, 31381059631, 94143360944, 282429536825, 847289140932

Offset

1,2

Links

Seiichi Manyama, Table of n, a(n) for n = 1..2096

Formula

a(n) ~ 3^(n-1). - Vaclav Kotesovec, Sep 11 2019

G.f.: Sum_{k>=1} phi(k) * x^k / (1 - 3*x^k). - Ilya Gutkovskiy, Feb 14 2020

a(n) = Sum_{k=1..n} 3^(gcd(k, n) - 1) = A054610(n)/3. - Seiichi Manyama, Apr 17 2021

a(n) = Sum_{k=1..n} 3^(n/gcd(n,k) - 1)*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 06 2021

Mathematica

Table[Sum[3^(n/d - 1)*EulerPhi[d], {d, Divisors[n]}], {n, 1, 30}] (* Vaclav Kotesovec, Sep 10 2019 *)

Prog

(PARI) a(n) = sum(k=1, n, 3^(gcd(k, n)-1)); \\ Seiichi Manyama, Apr 17 2021
(PARI) a(n) = sumdiv(n, d, eulerphi(n/d)*3^(d-1)); \\ Seiichi Manyama, Apr 17 2021
(PARI) my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k/(1-3*x^k))) \\ Seiichi Manyama, Apr 17 2021

Crossrefs

Cf. A000010, A001867, A034738, A054610.

Sequence in context: A155962 A027153 A319918 * A345029 A268744 A038747

Adjacent sequences: A034751 A034752 A034753 * A034755 A034756 A034757

Keyword

nonn

Author

Erich Friedman

Status

approved